At a receiver in a typical communication system, an analog-to-digital converter is utilized to convert a received continuous time signal into a discrete-time format. One problem which is encountered in this type of system is that the local receiver clock and the remote transmitter clock are asynchronous. If the receiver clock is slower than the transmitter clock, after a long enough period of time, one sample of the received continuous time signal will be lost. On the other hand, if the local receiver clock is faster than the remote transmitter clock, after a long enough period of time, an extra sample of the received continuous time signal will be obtained. The problem of synchronizing the local receiver clock to the remote transmitter clock is an important problem in many telecommunication systems. The present invention solves this problem in a communication system which utilizes pulse amplitude modulation.
PAM is discussed in some detail below to facilitate the understanding of the present invention. First baseband PAM is discussed and then passband PAM is discussed.
Baseband PAM is commonly used for metallic media, such as twisted wire pairs, where the signal spectrum is allowed to extend down to zero frequency (d.c.).
The PAM system is extended to passband transmission by introducing a sinusoidal carrier signal. Passband PAM is commonly used with transmission media with a highly constrained bandwidth such as radio. It uses two sinusoidal carrier signals of the same frequency, but with a ninety degree phase difference,which are modulated by the real and imaginary parts of a complex-valued baseband signal. Special cases of passband PAM are the commonly used phase-shift keying (PSK), amplitude and phase modulation (AM-PM) and quadrature amplitude modulation (QAM). See, for example, Edward A. Lee and David G. Mesershmitt, "Digital Communication", 1988, Kluwer Academic Publishers, Boston (hereinafter "Lee et al.") page 146.
A baseband communication system is shown in FIG. 1 (see Lee et al, page 148). The baseband system 10 of FIG. 1 includes a transmitter 12 and a receiver 14. At the transmitter 12, an incoming stream of bits B.sub.k is coded by a coder 16 into a stream of symbols A.sub.k.
While a bit can only assume the values of "0" or "1", a symbol assumes values from a predetermined alphabet of symbols. The alphabet is the set of symbols that are available for transmission. A baseband signal has a real-valued alphabet that is simply a list of real numbers, for example A.sub.k assumes values from the alphabet {-3, -1, +1, +3}. The coder 16 in the baseband transmitter 12 can, for example, map pairs of bits from the set {00, 01, 10, 11} into one of four levels from the alphabet {-3, -1, 1, 3}.
Because the coder 16 may map multiple bits into a single data symbol, there is a distinction between the "symbol rate" and the "bit rate". The "symbol rate" is also called the "baud rate". If for example, the coder maps two bits into each symbol, the symbol rate is one-half of the bit rate.
After being generated by the coder 16, the symbols A.sub.k are applied to a transmit filter 18. The transmit filter 18 produces a continuous-time signal s(t) for transmission over the continuous time channel 20.
The impulse response g(t) of the transmit filter 18 is called the pulse shape. The output s(t) of the transmit filter 18 is the convolution of the pulse shape and symbol sequence ##EQU1## where f.sub.b =1/T.sub.b is the baud rate. This signal can be interpreted as a sequence of possibly overlapped pulses with the amplitude of each pulse determined by a symbol. Such signals are termed pulse amplitude modulated (PAM) signals regardless of the pulse shape.
Baseband PAM and its generalization to passband are among the most common signaling methods. A variety of techniques including QAM, PSK, BPSK, PRK, QPSK, DPSK and AM-PM are special cases of PAM. See Lee et al, pages 149-150.
At the receiver 14, the signal R(t) which is received via the channel 20 is processed by a receiver filter 21 with an impulse response f(t) to produce the output signal Q(t). The received signal R(t) is also processed by a timing recovery circuit 22 which recovers a clock for use by the sampler 24. The signal Q(t) is sampled by the sampler circuit 24. The samples Q.sub.k outputted by the sampler 24 are processed by a decision device 26 to form the reconstructed symbols A.sub.k. Then the symbols are decoded by a decoder 28 to form the reconstructed bit stream B.sub.k.
Few practical communication channels can transmit baseband signals. Most physical transmission media are incapable of transmitting frequencies at d.c. and near d.c. which are contained in baseband signals.
The passband strategy is now considered. The passband strategy is considered using a discrete time signal representation.
The passband strategy is to construct a signal s(nT) where s(nT) is the discrete time analog to s(t) and where 1/T is the sampling rate. The signal s(nT) is in this case complex and contains information in both its real and imaginary parts. The signal s(nT) is modulated using a complex-valued carrier to obtain a modulated signal z(nT). A transmitter 30 for accomplishing this is schematically illustrated in FIG. 2 (see Lee et al, page 170). In the transmitter 30, the bits b.sub.k are processed by a coder 32 to produce the complex symbols a.sub.k =a(kT) where f.sub.s =1/T is the sampling frequency. The passband PAM signal has an alphabet that is a list of complex numbers, for example, {-1, -j, +1, +j}. For an alphabet with M=4 symbols, each symbol can represent log.sub.2 M=2 bits. A complex-valued alphabet is best described by plotting the alphabet as a set of points in a complex plane. Such a plot is called a signal constellation. Two popular constellations are illustrated in FIG. 3A and FIG. 3B. Returning now to FIG. 2, the complex symbols a(kT) are processed by the transmit filter 34. In a discrete-time representation, the transmit filter 34 of FIG. 2 has a transfer function h(nT). The output of the transmit filter 34 is the complex baseband signal ##EQU2## The complex baseband signal s(nT) is then multiplied in the modulator 38 by a complex valued carrier: EQU e.sup.i.omega.cnT =cos (.omega..sub.c nT)+j sin (.omega..sub.c nT)(3)
where .omega..sub.c is the carrier frequency. The modulator 38 thus outputs the complex modulated signal EQU z(nT)=s(nT)e.sup.i.omega.cnT ( 4)
The signal z(nT) is complex valued, so that it cannot be transmitted over a real-valued channel. However, all of the signal information is contained in the real part of the signal, which can be transmitted over a real-valued channel. Thus the circuit 39 is used to obtain x(nT)=.sqroot.2Rez(nT). The .sqroot.2 factor insures that the power of x(nT) is the same as the power in s(nT).
Timing in the discrete-time transmitter 30 of FIG. 2 is controlled by a transmitter clock (not shown).
The signal x(nT) is processed by the digital-to-analog converter 40 to convert to a continuous-time signal x(t) for transmission to a remote receiver via a transmission channel 41. It should be noted that the function z(nT) is analytic, i.e., its Fourier transform contains no negative frequency components.
The signal x(nT) may be represented as ##EQU3##
This indicates that the signal x(nT) is equal to two real-valued baseband PAM signals EQU s.sub.r (nT)=.sqroot.2 .SIGMA.Re a(kT)h((n-k)T) (6a) EQU s.sub.i (nT)=.sqroot.2 .SIGMA.Im a(kT)h((n-K)T) (6b)
modulated by the carrier signals cos (.omega..sub.c nT) and -sin (.omega..sub.c nT) respectively. These two carriers are 90 degrees out of phase with one another so that they are said to be in quadrature, where cos (.omega..sub.c nT) is called the in-phase component and sin (.omega..sub.c nT) is called the quadrature component.
FIG. 4 shows an alternative embodiment for the transmitter shown in FIG. 2 (see Lee et al, page 172). In the transmitter 50 of FIG. 4, bit stream b.sub.k is processed by a coder 51 to form the real symbol streams Re{a.sub.k } and Im{a.sub.k }. Each symbol stream is processed by a transmit filter 52 with a transfer function .sqroot.2h(nT) to form the first and second baseband signals s.sub.r (nT) and s.sub.i (nT). The first baseband signal is then multiplied by the carrier cos (.omega..sub.c nT) in the modulator 53 and the second baseband signal is multiplied by the carrier sin .omega..sub.c nT in the modulator 54. The signals are then combined using the subtractor element 55 to form the passband signal x(nT). This signal is converted to the analog signal x(t) by the digital-to-analog converter 56.
In an illustrative example, the sample rate f.sub.s =1/T=7200 Hz. The symbol rate f.sub.b =1/T.sub.b is illustratively 2400 symbols per sec. The bit rate may be 9600 bits/sec or 4800 bits/sec. The carrier frequency f.sub.c =.omega..sub.c /2.pi. is illustratively 1800 Hz.
It is an object of the present invention to provide a receiver for receiving the signal x(t) after it is transmitted to a remote location via a transmission channel. More particularly, it is an object to provide such a receiver which operates utilizing digital signal processing. In such a case it is necessary to sample the continuous-time signal which is transmitted via the channel to the receiver. Because the local clock which samples the received signal at the receiver may not be synchronized with the transmitter clock, the receiver must include some technique for local clock synchronization. It is a further object of the present invention to provide such a local clock synchronization technique.